MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climsubc1 Structured version   Unicode version

Theorem climsubc1 12431
Description: Limit of a constant  C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
Hypotheses
Ref Expression
climadd.1  |-  Z  =  ( ZZ>= `  M )
climadd.2  |-  ( ph  ->  M  e.  ZZ )
climadd.4  |-  ( ph  ->  F  ~~>  A )
climaddc1.5  |-  ( ph  ->  C  e.  CC )
climaddc1.6  |-  ( ph  ->  G  e.  W )
climaddc1.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climsubc1.h  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `
 k )  -  C ) )
Assertion
Ref Expression
climsubc1  |-  ( ph  ->  G  ~~>  ( A  -  C ) )
Distinct variable groups:    C, k    k, F    ph, k    A, k   
k, G    k, M    k, Z
Allowed substitution hint:    W( k)

Proof of Theorem climsubc1
StepHypRef Expression
1 climadd.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climadd.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climadd.4 . 2  |-  ( ph  ->  F  ~~>  A )
4 climaddc1.6 . 2  |-  ( ph  ->  G  e.  W )
5 climaddc1.5 . . 3  |-  ( ph  ->  C  e.  CC )
6 0z 10293 . . 3  |-  0  e.  ZZ
7 uzssz 10505 . . . 4  |-  ( ZZ>= ` 
0 )  C_  ZZ
8 zex 10291 . . . 4  |-  ZZ  e.  _V
97, 8climconst2 12342 . . 3  |-  ( ( C  e.  CC  /\  0  e.  ZZ )  ->  ( ZZ  X.  { C } )  ~~>  C )
105, 6, 9sylancl 644 . 2  |-  ( ph  ->  ( ZZ  X.  { C } )  ~~>  C )
11 climaddc1.7 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
12 eluzelz 10496 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
1312, 1eleq2s 2528 . . . 4  |-  ( k  e.  Z  ->  k  e.  ZZ )
14 fvconst2g 5945 . . . 4  |-  ( ( C  e.  CC  /\  k  e.  ZZ )  ->  ( ( ZZ  X.  { C } ) `  k )  =  C )
155, 13, 14syl2an 464 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ZZ  X.  { C } ) `  k
)  =  C )
165adantr 452 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  C  e.  CC )
1715, 16eqeltrd 2510 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ZZ  X.  { C } ) `  k
)  e.  CC )
18 climsubc1.h . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `
 k )  -  C ) )
1915oveq2d 6097 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  -  ( ( ZZ  X.  { C } ) `  k
) )  =  ( ( F `  k
)  -  C ) )
2018, 19eqtr4d 2471 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `
 k )  -  ( ( ZZ  X.  { C } ) `  k ) ) )
211, 2, 3, 4, 10, 11, 17, 20climsub 12427 1  |-  ( ph  ->  G  ~~>  ( A  -  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990    - cmin 9291   ZZcz 10282   ZZ>=cuz 10488    ~~> cli 12278
This theorem is referenced by:  clim2ser  12448  ulmdvlem1  20316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282
  Copyright terms: Public domain W3C validator